from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2200, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,17,8]))
pari: [g,chi] = znchar(Mod(247,2200))
Basic properties
Modulus: | \(2200\) | |
Conductor: | \(1100\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1100}(247,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2200.gi
\(\chi_{2200}(247,\cdot)\) \(\chi_{2200}(383,\cdot)\) \(\chi_{2200}(487,\cdot)\) \(\chi_{2200}(663,\cdot)\) \(\chi_{2200}(1303,\cdot)\) \(\chi_{2200}(1423,\cdot)\) \(\chi_{2200}(1967,\cdot)\) \(\chi_{2200}(2127,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
Field of values: | \(\Q(\zeta_{20})\) |
Fixed field: | 20.20.140227447093420901489257812500000000000000000000.2 |
Values on generators
\((551,1101,177,1201)\) → \((-1,1,e\left(\frac{17}{20}\right),e\left(\frac{2}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 2200 }(247, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)